why is it that some amino acids allow alpha helix formation in a protein structure? For instance, I read that Alanine, Glu, Leu, and Met are good for forming alpha helixes while a few others are poor at it. What property do those four amino acids share?

same here.to form an alpha helix alot of hydrogen bonds are needed the smaller the molecules they are the smaller and stabler the helix will be. just imagine the helic. in the chain in the secondary structure the start of the molecule will be hydrogen bonded to it's end. just take a wire and twist it in different loops of different diameters and check their stabilities.smaller the better.